Solutions of linear equations in pseudorandom sets

A common theme in modern combinatorics consists in proving sparse analogues of results known in the dense setting. We review some of these for linear systems of equations. We first prove sparse analogues for random sets of Szemerédi s theorem and Rado s theorem via the hypergraph container method. Finally, we prove a sparse analogue for quasirandom sets of Roth s theorem via the regularity method